WEBVTT
Kind: captions
Language: en
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so welcome to the lecture series on mathematical
methods and their applications uh in the last
00:00:25.070 --> 00:00:30.490
lecture in the last few lectures we have seen
that what laplace transforms are what are
00:00:30.490 --> 00:00:37.000
their properties and how can we solve an ordinary
differential equation or a partial differential
00:00:37.000 --> 00:00:44.850
equation ontical equations using laplace transforms
now the next topic is uh fourier series now
00:00:44.850 --> 00:00:55.019
in this uh lectures we will see that what
fourier series are and uh what is the convergence
00:00:55.019 --> 00:01:01.080
theorem for fourier series and what are the
applications of fourier series or fourier
00:01:01.080 --> 00:01:10.200
integrals ok now uh first few introduction
first introduction on uh fourier series it
00:01:10.200 --> 00:01:15.270
is named after a french mathematician and
a physicist jacques fourier who was the first
00:01:15.270 --> 00:01:21.689
to use fourier series in his work a series
expansion of a function in terms of a trigonometric
00:01:21.689 --> 00:01:26.320
functions cos m x and sin n x is called a
fourier series so fourier series is nothing
00:01:26.320 --> 00:01:32.600
but if you express a function in terms of
sine and cosines then we call such a series
00:01:32.600 --> 00:01:39.329
as fourier series many functions including
some discontinuous periodic functions can
00:01:39.329 --> 00:01:46.109
be written in the fourier series and hence
it has a wide application in solving ordinary
00:01:46.109 --> 00:01:51.659
and partial differential equations
now let us start with this set set is one
00:01:51.659 --> 00:02:00.210
cos pi x by l cos two pi x by l and so on
sine pi x by l sine two pi x by l and so on
00:02:00.210 --> 00:02:05.990
now what is an orthogonal set of functions
how do we define it suppose we have some functions
00:02:05.990 --> 00:02:13.950
suppose functions are f one f two f three
and so on and it is defined suppose defined
00:02:13.950 --> 00:02:25.980
in a comma b then we say that this set of
function is orthogonal if in this interval
00:02:25.980 --> 00:02:41.260
if integral a to b f i x into f j x d x is
equals to zero for all i not equal to j so
00:02:41.260 --> 00:02:47.870
if this condition hold if this condition hold
for set of functions f one f two up to uh
00:02:47.870 --> 00:02:52.319
f three and so on which is defined in the
interval a comma b then we say that the set
00:02:52.319 --> 00:03:02.010
of functions are orthogonal ok now consider
this set one cos pi x by l cos two pi x by
00:03:02.010 --> 00:03:09.019
l and so on sin pi x by l sin two pi x by
l and so on now in the interval minus l to
00:03:09.019 --> 00:03:17.159
l if we see if we carefully see then minus
integral minus l to l cos m pi x by l d x
00:03:17.159 --> 00:03:23.060
and this equal to zero again the second inequality
holds the second inequality also hold it is
00:03:23.060 --> 00:03:31.250
zero when m not equal to n the next inequality
minus l to l sin m pi x by l and sin n pi
00:03:31.250 --> 00:03:37.659
x by l d x equal to zero when m not equal
to n and is l when m equal to n so this we
00:03:37.659 --> 00:03:45.359
can derive very easily and minus l to l cos
pi m pi x by l into this is also zero for
00:03:45.359 --> 00:03:50.750
all m and n
so hence if we take any two multiples in this
00:03:50.750 --> 00:03:56.459
set in this set if we take any two different
functions and integrate from minus l to l
00:03:56.459 --> 00:04:04.090
it is always zero so we say that this set
of this set is an orthogonal set of functions
00:04:04.090 --> 00:04:09.200
ok because it satisfies this condition it
satisfies this condition because integral
00:04:09.200 --> 00:04:17.660
a to b f i x f j x is zero for all i not equal
to j and this set satisfies this property
00:04:17.660 --> 00:04:24.920
hence we say that this set is nothing but
an orthogonal set of functions now uh now
00:04:24.920 --> 00:04:31.820
let f x be a periodic function of period two
n and define an interval minus l to l ok assume
00:04:31.820 --> 00:04:38.319
that now we are assuming that it is it can
be expressed as the linear combination of
00:04:38.319 --> 00:04:45.810
trigonometric functions of cos m x and sin
m x so what i want to say so first i construct
00:04:45.810 --> 00:04:54.250
a orthogonal set of functions and that set
is nothing but this set one cos pi x by l
00:04:54.250 --> 00:05:11.650
cos two pi x by l and so on sin pi x by l
sin two pi x by l and so on ok this is an
00:05:11.650 --> 00:05:19.680
orthogonal set of function and a function
which is periodic ok in the interval minus
00:05:19.680 --> 00:05:27.419
l having a period two l defined on minus l
to l which is given can be expressed as a
00:05:27.419 --> 00:05:33.940
linear combination of these functions that
is some multiple of this we are taking for
00:05:33.940 --> 00:05:41.750
convenience as a naught by two plus a one
into this a two into this and so on b one
00:05:41.750 --> 00:05:47.780
into this b two into this and so on so which
can be written as summation n from one to
00:05:47.780 --> 00:06:04.479
infinity a n cos n pi x by l plus summation
n from one to infinity b n sin n pi x by l
00:06:04.479 --> 00:06:08.919
ok
this function we can write as a linear combination
00:06:08.919 --> 00:06:18.699
of these trigonometric functions sine and
cos now if you want to find out the values
00:06:18.699 --> 00:06:25.410
of a naught a n b n and so on so how can you
find these values so what is this expression
00:06:25.410 --> 00:06:37.610
basically this is a naught by two plus a one
cos pi x by l plus a two cos two pi x by l
00:06:37.610 --> 00:06:49.139
and so on plus a n cos n pi x by l and so
on this is what we are having here and here
00:06:49.139 --> 00:07:02.680
it is b one sin pi x by l plus b two sin pi
x sin two pi x by l sin two pi x by l and
00:07:02.680 --> 00:07:12.879
so on plus b n sin n pi x by l and so on so
basically the uh the linear combination of
00:07:12.879 --> 00:07:20.150
these functions is this thing or this thing
ok now to find out the value of a naught let
00:07:20.150 --> 00:07:26.810
us integrate both the sides from minus l to
plus l ok so if we integrate from minus l
00:07:26.810 --> 00:07:34.060
to l in this side it is minus l to l f x d
x is equals to a naught by two integral minus
00:07:34.060 --> 00:07:44.169
l to plus l d x plus now integral minus l
to l cos n pi x by l for every n is zero and
00:07:44.169 --> 00:07:51.550
integral minus l to l sine n pi x by l d x
is zero for any n so that means these all
00:07:51.550 --> 00:08:02.699
are zero so this implies a naught is nothing
but one by l times minus l to l f x d x
00:08:02.699 --> 00:08:13.470
so that is how we can compute a naught one
by l minus integral minus l to l f x d x ok
00:08:13.470 --> 00:08:24.240
now suppose you want to compute a n a n uh
n may be one two three and so on suppose you
00:08:24.240 --> 00:08:34.360
want to compute this a n ok so you multiply
both sides by cos n pi x by l ok so when you
00:08:34.360 --> 00:08:43.320
multiply both sides by cos pi x by l it is
f x into cos n pi x by l integrate from minus
00:08:43.320 --> 00:08:51.830
l to plus l d x is equal to now here if you
multiply and integrate from minus l to l cos
00:08:51.830 --> 00:09:03.880
pi x by l is zero for any n now if it will
be not zero cos into cos is not equal to zero
00:09:03.880 --> 00:09:10.540
only when m equal to n that is n equal to
n and all in all other cases it will be zero
00:09:10.540 --> 00:09:17.380
and cos and sine multiple is always zero for
any m and n this is always zero here what
00:09:17.380 --> 00:09:25.360
we have here it is nothing but a n integral
minus l to l cos square n pi x by l into d
00:09:25.360 --> 00:09:33.720
x and we have already seen that this value
is nothing but l a n into l because when m
00:09:33.720 --> 00:09:39.300
equal to n so this value is nothing but l
this we have already seen so from here we
00:09:39.300 --> 00:09:49.130
can say that a n is nothing but one by l times
integral minus l to l f x cos n pi x by l
00:09:49.130 --> 00:09:57.710
into d x so this is how we can find out a
n and what is n n may be one may be two may
00:09:57.710 --> 00:10:06.990
be three and so on ok clear
now uh now suppose you want to find out b
00:10:06.990 --> 00:10:16.959
n in the same way you multiply both the sides
by sin n pi x by l this is integral minus
00:10:16.959 --> 00:10:27.339
l to l f x into sin n pi x by l into d x integrate
both the sides here integral minus l to l
00:10:27.339 --> 00:10:34.370
sin n pi x by l is zero and the multiple of
sine and cos for any m and n integral minus
00:10:34.370 --> 00:10:41.029
l to l is always zero so all these terms are
zero and here we have and here when m is not
00:10:41.029 --> 00:10:50.690
equal to n it is zero we left with only b
n so integral minus l to l b n into sin square
00:10:50.690 --> 00:11:00.670
n pi x by l into d x and this value is nothing
but l when m equal to n so it is b n into
00:11:00.670 --> 00:11:12.760
l so this implies b n is nothing but one by
l integral minus l to l f x sin n pi x by
00:11:12.760 --> 00:11:22.990
l into d x so these formulas are called basically
eulers formula ok to find a naught a n and
00:11:22.990 --> 00:11:30.829
b n so uh if we assume that if we express
function periodic function f x as a linear
00:11:30.829 --> 00:11:37.860
combination of cos trigonometric functions
of cos and sine then a naught a n and b n
00:11:37.860 --> 00:11:47.209
can be find using these expressions ok
now let us solve some problems based on this
00:11:47.209 --> 00:11:56.860
that how we can find fourier series of periodic
function so basically uh so basically if we
00:11:56.860 --> 00:12:02.980
have a periodic function f x of period two
l defined in minus l to l so what will be
00:12:02.980 --> 00:12:09.020
f x what we have learnt that f x is nothing
but a naught by two plus summation n varying
00:12:09.020 --> 00:12:20.399
from one to infinity a n cos n pi x by l plus
summation n from one to infinity b n sin n
00:12:20.399 --> 00:12:28.410
pi x by l now what are what is a naught a
naught is nothing but one by l integral minus
00:12:28.410 --> 00:12:36.899
l to l f x d x this we have derived a n is
nothing but one by l integral minus l to l
00:12:36.899 --> 00:12:49.010
f x cos n pi x by l into d x and what is b
n it is one by l integral minus l to l f x
00:12:49.010 --> 00:13:00.360
sin n pi x by l into d x here in these two
expressions n is one two three and so on so
00:13:00.360 --> 00:13:06.880
now let us solve this problem here it is given
that function is periodic and has a period
00:13:06.880 --> 00:13:14.431
two l oh sorry two pi ok and what is a what
is function what is uh nature of function
00:13:14.431 --> 00:13:23.240
function is pi plus x when x is varying from
zero to zero to pi zero to minus pi and it
00:13:23.240 --> 00:13:34.000
is zero when zero less than equal to x less
than pi and f x plus two pi is f pi f x because
00:13:34.000 --> 00:13:43.940
period is two pi so what is the shape of the
function when it is minus pi so it is zero
00:13:43.940 --> 00:13:50.680
and when it is zero it is pi so here it is
like this and from zero to pi suppose pi is
00:13:50.680 --> 00:13:58.649
here from zero to pi it is zero
again from pi to two pi it is this function
00:13:58.649 --> 00:14:08.920
say i am having the same height and from pi
to three pi is again zero then three pi to
00:14:08.920 --> 00:14:16.160
four pi again it is uh this line and four
pi to five pi it is zero so this function
00:14:16.160 --> 00:14:21.279
is something like this is the shape of this
i mean graph of this function now how can
00:14:21.279 --> 00:14:31.410
we find the fourier series of this function
so so we will express this function in this
00:14:31.410 --> 00:14:39.310
form a naught by two plus summation a n cos
n pi x by l plus b n plus summation b n sin
00:14:39.310 --> 00:14:47.970
n pi x by l now we first compute a naught
and a n b n what is a naught a naught is nothing
00:14:47.970 --> 00:14:56.240
but one by l now instead of l here we have
pi because two l is two pi oh that means l
00:14:56.240 --> 00:15:05.170
equal to pi it is one by pi integral minus
pi to plus pi f x d x so this is nothing but
00:15:05.170 --> 00:15:12.860
one by pi integral from minus pi to zero it
is pi plus x otherwise it is zero [vocalized-noise]
00:15:12.860 --> 00:15:21.040
so it is nothing but one by pi and this term
is nothing but pi x plus x square upon two
00:15:21.040 --> 00:15:27.540
minus pi to zero upper limit minus lower limit
when you apply the lower limit it is nothing
00:15:27.540 --> 00:15:39.399
but negative of pi into minus pi plus pi square
upon two and it is minus pi square by two
00:15:39.399 --> 00:15:45.930
that is pi by two
so this is a naught ok so a naught for this
00:15:45.930 --> 00:15:51.220
problem is pi by two so i am writing here
a naught is nothing but pi by two for this
00:15:51.220 --> 00:16:02.550
problem ok so a naught is pi by two now let
us find a n a n a n is nothing but one by
00:16:02.550 --> 00:16:15.089
pi integral minus pi to plus pi f x and it
is cos n pi x by l l is pi into d x so pi
00:16:15.089 --> 00:16:26.339
pi cancel out it is nothing but one by pi
integral minus pi to zero pi plus x cos n
00:16:26.339 --> 00:16:33.449
x d x which is nothing but one by pi now you
will integrate this apply integration by parts
00:16:33.449 --> 00:16:43.760
so it is first as it is integration of second
sin n x upon n minus this derivative derivative
00:16:43.760 --> 00:16:55.490
is one into integration of this that is minus
cos n x upon n square from minus pi to zero
00:16:55.490 --> 00:17:04.620
so this is nothing but one by pi when uh when
x equal to zero sine is zero and when x equal
00:17:04.620 --> 00:17:10.670
to pi or minus pi sin n pi is zero so this
term is zero from the lower limit and the
00:17:10.670 --> 00:17:18.540
upper limit so this term is zero when x is
zero cos zero is one and when x is minus pi
00:17:18.540 --> 00:17:28.980
it is cos n pi and cos n pi is uh cos n pi
is minus one for negative n for odd n i mean
00:17:28.980 --> 00:17:37.860
what is cos n pi cos n pi is nothing but minus
one k to the power n when n is odd it is minus
00:17:37.860 --> 00:17:46.290
one when n is even it is one so uh this is
nothing but minus minus plus it is one by
00:17:46.290 --> 00:17:56.790
n square minus cos n pi by n square so this
is nothing but one by pi n square one minus
00:17:56.790 --> 00:18:03.890
minus one k to the power n so this is this
is a n for this problem so what is a n a n
00:18:03.890 --> 00:18:12.600
will be nothing but one by n square pi one
minus minus one k to the power n
00:18:12.600 --> 00:18:18.030
now let us compute the last term that is b
n b n will be nothing but again one by pi
00:18:18.030 --> 00:18:31.660
integral minus pi to pi it is f x sin n pi
x by l l is pi so this is one by pi again
00:18:31.660 --> 00:18:41.510
integral minus pi to zero it is pi plus x
by the definition of the function now again
00:18:41.510 --> 00:18:49.910
you will integrate by parts here pi plus x
sin n x ok integration of this so integration
00:18:49.910 --> 00:18:58.380
of this is nothing but minus cos n x by n
minus derivative of first integration of second
00:18:58.380 --> 00:19:08.840
that is minus sin n x by n square from minus
pi to zero now again this term when you take
00:19:08.840 --> 00:19:14.820
x equal to zero it is zero when you take x
equal to minus pi it is zero and here you
00:19:14.820 --> 00:19:22.340
take you can take minus outside one by pi
when you substitute x as minus pi it is zero
00:19:22.340 --> 00:19:34.960
when you take x equal to zero it is uh it
is pi by n ok when you take when you take
00:19:34.960 --> 00:19:42.450
x equal to zero it is it is pi pi by n and
when you take x equal to minus pi it is zero
00:19:42.450 --> 00:19:52.200
so it is nothing but minus one by n ok so
what will be so b n is nothing but minus one
00:19:52.200 --> 00:20:02.170
by n so what will be the fourier series of
this function
00:20:02.170 --> 00:20:08.130
so fourier series representation of this function
f x is nothing but f x can be written as a
00:20:08.130 --> 00:20:16.160
naught by two what is a naught a naught is
pi by two a naught by two that is pi by four
00:20:16.160 --> 00:20:24.670
pi by four plus summation n from one to infinity
a n a n is nothing but this expression so
00:20:24.670 --> 00:20:39.110
it is one by n square pi one minus minus one
k to the power n into cos n pi x by l plus
00:20:39.110 --> 00:20:47.120
summation n from one to infinity it is b n
sin n pi x by l what is b n minus one by n
00:20:47.120 --> 00:20:57.440
it is minus one by n sin n pi x by l so which
if you expand this it is nothing but pi by
00:20:57.440 --> 00:21:10.180
four plus when n is minus when n is one it
is nothing but zero when n is two ok when
00:21:10.180 --> 00:21:19.130
n is one it is nothing but two for odd for
even values of n it is zero so that means
00:21:19.130 --> 00:21:27.930
from here only odd values come it is nothing
but two will be outside two upon pi and it
00:21:27.930 --> 00:21:36.940
is nothing but and this l is nothing but pi
sorry this l is nothing but pi pi pi cancels
00:21:36.940 --> 00:21:47.560
out because l is nothing but pi so you substitute
when you substitute x as one so when x is
00:21:47.560 --> 00:21:53.540
one it is two so two i am taking outside pi
i am taking outside [FL] it is cos x upon
00:21:53.540 --> 00:22:08.100
one square plus cos three square and so on
and minus it is uh sin x upon one plus sin
00:22:08.100 --> 00:22:18.950
two x upon two plus sin three x upon three
so this will be the fourier series representation
00:22:18.950 --> 00:22:27.220
of this function f x ok
now let us solve one more problem based on
00:22:27.220 --> 00:22:34.480
this uh fourier series representation of this
function and hence reduce this series ok so
00:22:34.480 --> 00:22:45.990
let us solve this problem also so here period
is again two pi so l equal to pi ok and function
00:22:45.990 --> 00:22:53.350
is x minus x square so directly compute a
naught a n and b n after calculating a naught
00:22:53.350 --> 00:22:58.900
a n and b n we substitute it over here find
f x and hence the fourier series representation
00:22:58.900 --> 00:23:06.040
of this function and then substituting some
values of x we will try to obtain this uh
00:23:06.040 --> 00:23:16.590
series ok so what is f x here f x is x minus
x square now what will be a naught again one
00:23:16.590 --> 00:23:28.080
by pi minus pi to plus pi f x d x it is one
by pi minus pi to plus pi x minus x square
00:23:28.080 --> 00:23:35.580
by two d x now this x is a odd function from
minus pi to pi it is zero and this x square
00:23:35.580 --> 00:23:40.790
is an even function so it will be two times
so we can easily write it is minus two by
00:23:40.790 --> 00:23:50.690
upon pi zero to pi x square d x ok and it
is nothing but minus two upon pi it is x cube
00:23:50.690 --> 00:24:00.040
upon three zero to pi so which is nothing
but minus two upon pi pi cube by three and
00:24:00.040 --> 00:24:08.300
it is equal to minus two by three pi square
so this is a naught for this problem so i
00:24:08.300 --> 00:24:15.090
am writing a naught over here so a naught
is minus two by three pi square now let us
00:24:15.090 --> 00:24:25.410
compute a n for this function so what will
be a n a n is one by pi integral minus pi
00:24:25.410 --> 00:24:43.100
to plus pi f x cos n pi x by l into d x ok
now again x into this is odd function it will
00:24:43.100 --> 00:24:48.100
be zero from minus pi to plus pi and x square
into this is an even function so it will be
00:24:48.100 --> 00:24:57.270
two times so we can write it here minus two
upon pi integral zero to pi x square cos n
00:24:57.270 --> 00:25:04.680
x d x so this is nothing but minus two upon
pi you will integrate by parts x square sine
00:25:04.680 --> 00:25:16.930
n x upon n minus two x this is minus cos n
x upon n square then uh plus two this is minus
00:25:16.930 --> 00:25:29.790
sin n x upon n cube and the whole expression
from zero to pi now whatever terms of sine
00:25:29.790 --> 00:25:36.110
from zero to pi is zero because upper limit
is zero sin n pi is zero and sin zero is zero
00:25:36.110 --> 00:25:42.630
this is zero this is zero now only this term
lefts now it is also when x is zero is zero
00:25:42.630 --> 00:25:53.160
so only the uh above limit left it is minus
two upon pi it is two pi and it is cos n pi
00:25:53.160 --> 00:26:03.220
upon n square which is nothing but pi pi cancel
out so it is minus four by n square into minus
00:26:03.220 --> 00:26:10.950
one k to the power n so a n is nothing but
minus four upon n square minus one k to the
00:26:10.950 --> 00:26:18.040
power n when n varying from one two three
and so on ok now b n can be calculated in
00:26:18.040 --> 00:26:25.890
the similar way so what will be b n b n will
be one by pi integral minus pi to plus pi
00:26:25.890 --> 00:26:40.230
x minus x square sin n pi x by l into d x
now x into this is an even function and this
00:26:40.230 --> 00:26:45.840
is an odd function this will be zero and we
left with only two upon pi integral zero to
00:26:45.840 --> 00:26:53.830
pi x sin n x d x
now again integration by parts it is first
00:26:53.830 --> 00:27:01.750
as it is integral second it is minus cos n
x upon n minus one into integral of this that
00:27:01.750 --> 00:27:12.510
is minus sin n x upon n square from zero to
pi now this will be zero when x is pi and
00:27:12.510 --> 00:27:18.570
when x is zero only this will be left so it
is minus two upon pi n i am taking outside
00:27:18.570 --> 00:27:29.111
it is only exists when x is pi so it is pi
into minus one k to the power n ok pi pi cancel
00:27:29.111 --> 00:27:38.580
out b n is nothing but b n is nothing but
uh minus two by n into minus one k to the
00:27:38.580 --> 00:27:50.160
power n ok now so what is a fourier series
representation of this function what is the
00:27:50.160 --> 00:27:58.390
function x minus x square so f x will be equal
to a naught by two a naught is is this term
00:27:58.390 --> 00:28:08.580
eta a naught by two is minus pi square upon
three plus summation n from one to infinity
00:28:08.580 --> 00:28:19.830
a n a n is minus four by n square minus one
k to the power n cos n x plus summation n
00:28:19.830 --> 00:28:26.070
from one to infinity b n b n is minus two
k to the power two upon n minus one k to the
00:28:26.070 --> 00:28:36.230
power n sin n x so what the series is this
is minus pi square upon three minus four times
00:28:36.230 --> 00:28:48.180
when you take n equal to one when n is one
it is it is minus cos x when n is two it is
00:28:48.180 --> 00:28:56.100
plus cos two x upon two square when n is three
it is minus cos three x upon three square
00:28:56.100 --> 00:29:06.990
and so on and it is minus two times when n
is one it is minus sin x upon one plus when
00:29:06.990 --> 00:29:13.010
n is two it is sin two x upon two when n is
three it is minus sin three x upon three and
00:29:13.010 --> 00:29:21.950
so on ok
so now we want to formulate this series ok
00:29:21.950 --> 00:29:26.130
this is the fourier representation of fourier
representation of this function ok the first
00:29:26.130 --> 00:29:36.370
part is over now we want to deduce this equal
two pi square by twelve now it is one by one
00:29:36.370 --> 00:29:42.270
square one by two square with alternate signs
ok one by three square one by four square
00:29:42.270 --> 00:29:48.650
with alternate sign and we dont want any term
from this series so this we can obtain putting
00:29:48.650 --> 00:29:53.780
x equal to zero when you put x equal to zero
all these terms are cos x cos two x cos three
00:29:53.780 --> 00:30:00.520
x all will be one and all these will be zero
so put x equal to zero so if we put x equal
00:30:00.520 --> 00:30:06.790
to zero the left hand side is zero it is minus
pi square by three take minus common from
00:30:06.790 --> 00:30:12.520
here it is four times it is one by one square
minus one by two square plus one by three
00:30:12.520 --> 00:30:19.810
square and so on and this is zero so what
will be the series of this it is nothing but
00:30:19.810 --> 00:30:26.230
one upon one square plus one by two square
plus one by three square and so on is equal
00:30:26.230 --> 00:30:35.040
to pi square by three will go here divide
by four so it is pi square by twelve so hence
00:30:35.040 --> 00:30:42.710
this result can be obtained ok so in the next
class we will see that what how we can say
00:30:42.710 --> 00:30:48.220
that what is the convergence theorem for the
fourier series that we will see ok so
00:30:48.220 --> 00:30:48.909
thank you very much